Classroom Resource

Substitution to Find Limits

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Finding limits for the vast majority of points for a given function is as simple as substituting the number that x approaches into the function. Since this turns evaluating limits into an algebra-level substitution, most questions involving limits focus on the cases where substituting does not work. How can you decide if substitution is an appropriate analytical tool for finding a limit?

This informational material will explain how to find a limit of a function using algebraic substitution and when this method is appropriate. There is a corresponding video available. Practice questions with a PDF answer key are provided.

Content Standards

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.7
Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

Unpacked Content

Vocabulary

Limit
Convergent
Divergent
Continuous function
Discontinuous function
Discontinuity (infinite, jump, removable)

Knowledge

Students know:

How to graph the families of functions.
How to factor, simplify, and rationalize functions.
How to evaluate functions at a given input value.

Skills

Students are able to:

Identify continuity and discontinuity (if discontinuous, identify the type).
Calculate limits.

Understanding

Students understand that:

Functions have various types of continuity and discontinuity.
That limits can be determined numerically, graphically, and algebraically.
converging means that a limit exists, while diverging means that a limit does not exist.

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.24
Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

Unpacked Content

Vocabulary

Function
Relative Maximum
Relative Minimum
Symmetry (Even/Odd)
End Behavior
Asymptotes
Intercepts
Increasing/Decreasing Intervals
Periodicity
Absolute Maximum
Absolute Minimum

Knowledge

Students know:

Properties of functions and make connections between different representations of the same function

Skills

Students are able to:

Compare properties of functions when represented in different ways (algebraically, graphically, numerically in tables or by verbal descriptions).

Understanding

Students understand that:

Each representation provides a unique perspective of the function.
Different representations are most appropriate for revealing certain key features of the function

Interdisciplinary Connection(s)

Link to Resource

https://www.ck12.org/c/calculus/evaluate-limits-using-substitution/lesson/Subst…

Content Type / Source

Resource Provider

Other

Resource Provider other

CK-12

Accessibility

License

License Type

CUSTOM

Custom License Type (URL)

https://www.ck12info.org/terms-of-use/

Comments or Notes

Students can create a free account online to save their progress.

Approved Date

2022-05-04

Curated by

habradley

Design by Adaptive Theme